Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

Figure 7.3.4 shows the transverse fields of the four lowest modes of a guide with $a = 2 b$ . They are drawn at the instant at which the transverse fields are largest in this cross-section, when $E_{z}$ or $H_{z}$ , which is $9 0^{\circ}$ out of phase with them, is zero there.

Transverse field patterns of the four lowest modes of a guide with a=2b: \vec E orange, \vec H dashed blue. TE_{10}: E_y\propto\sin\frac{\pi x}{a}, H_x\propto-\sin\frac{\pi x}{a}, nothing varies with y. TE_{20}: two TE_{10} patterns side by side, opposite in sign. TE_{11}: \vec E follows the contours of H_z=\cos\frac{\pi x}{a}\cos\frac{\pi y}{b} round the corners and \vec H runs from corner to corner. TM_{11}: \vec E radiates from the centre, where E_z is largest, to the walls, and \vec H circles round it on the contours of E_z. — **Figure 7.3.4.** Transverse field patterns of the four lowest modes of a guide with $a = 2 b$ : $E$ orange, $H$ dashed blue. TE $_{10}$ : $E_{y} \propto sin \frac{π x}{a}$ , $H_{x} \propto - sin \frac{π x}{a}$ , nothing varies with $y$ . TE $_{20}$ : two TE $_{10}$ patterns side by side, opposite in sign. TE $_{11}$ : $E$ follows the contours of $H_{z} = cos \frac{π x}{a} cos \frac{π y}{b}$ round the corners and $H$ runs from corner to corner. TM $_{11}$ : $E$ radiates from the centre, where $E_{z}$ is largest, to the walls, and $H$ circles round it on the contours of $E_{z}$ .

How the pictures are made

For a TE mode Eq. (7.3.21) gives $E_{T} \propto a_{z} \times \nabla_{T} H_{z}$ and $H_{T} \propto \nabla_{T} H_{z}$ : the lines of $E$ are the contours of $H_{z}$ and the lines of $H$ cross them at right angles. For a TM mode Eq. (7.3.8) gives $E_{T} \propto \nabla_{T} E_{z}$ and $H_{T} \propto a_{z} \times \nabla_{T} E_{z}$ : now the lines of $H$ are the contours of $E_{z}$ . The contours are drawn at equal steps of level, so that their density follows the field strength. In every picture $E$ meets the walls at right angles and $H$ runs along them, the two boundary conditions of a perfect conductor. The index $m$ ( $n$ ) counts the half-period variations along $x$ ( $y$ ).

The tool below shows the $6 \times 3 cm$ guide of Figure 7.3.3 at $6 GHz$ , starting with TE $_{11}$ . Tap the bars of the mode chart to see each pattern, and move $f$ into the green single-mode band, $2.5$ to $5 GHz$ .

InteractiveLoading simulation…

Quick check

Looking at Figure 7.3.4, which of the four modes are excited by a thin probe parallel to $y$ that spans the height of the guide at $x = a /2$ ?

The probe couples to $E_{y}$ along the line $x = a /2$ . TE $_{10}$ has its largest $E_{y}$ there and is excited. TE $_{20}$ has $E_{y} = 0$ on the centre line and is not. TE $_{11}$ and TM $_{11}$ both have $E_{y} \propto sin \frac{π x}{a} cos \frac{π y}{b}$ , which changes sign at $y = b /2$ : the two halves of the probe see equal and opposite fields and the net coupling is zero. A symmetric probe in the middle of the broad side launches a clean TE $_{10}$ wave.

7.3.5 Field patterns in the cross-section